# Stability Graph Method

## History

Empirical databases such as the Q and RMR systems were develop as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method dimensions each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories.

Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N' to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings.

The database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions

## Input Parameters

### Modified Stability Number, N'

N' = Q' x A x B x C

where:

Q = RQD/Jn x Jr/Ja

RQD/Jn: is the measure of block size for a jointed rock mass Jr/Ja: is the measure of joint surface strength and stiffness

A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.

B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability

C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability

### Shape factor, S

The shape factor S also known as the hydraulic radius (HR) is defined by the ratio of a stope free face's area to perimeter, calculated as:

HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)

## Calculation of Input Parameters

### Q'

The input parameters for Q' can be measured and calculated as described in Site investigation and rock mass characterization

### Rock Stress Factor A

Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience Phase 2.0. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress the measurement needs to be taken at the center of the free stope face being measured, as seen in Figure 2.

Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis

Figure 2: Determining max induced compressive stress using Phase 2.0

### Joint Orientation Factor, B

Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's DIPS. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor will be used in calculating the stability number. This is done because the joint set with the lowest B factor will be the comprising joint set, and be the most likely mode of failure in the stope.

Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis

Figure 4: Measuring the angle between two planes using DIPS

Direct Calculation of Interplane Angle

Given the trend and plunge of the pole vector of the stope face, w, and a joint set,j, the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):

For the stope face:

For the joint plane:

Next calculate the dot product, $\displaystyle w\cdotj$ , between the stope face and the joint set

$\displaystyle w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j$

The true interplane angle, , is given by:

$\displaystyle \alpha = cos^-1(w\cdotj)$